Purely leptonic decays of the ground charged vector mesons

The study of the purely leptonic decays of the ground charged vector mesons is very interesting and significant in determining the CKM matrix elements, obtaining the decay constant of vector mesons, examining the lepton flavor universality, and searching for new physics beyond the standard model. These purely leptonic decays of the ground charged vector mesons are induced by the weak interactions within the standard model, and usually have very small branching ratios, ${\cal B}({\rho}^{-}{\to}{\ell}^{-}{\nu}_{\ell})$ ${\sim}$ ${\cal O}(10^{-13})$, ${\cal B}(K^{{\ast}-}{\to}{\ell}^{-}{\nu}_{\ell})$ ${\sim}$ ${\cal O}(10^{-13})$, ${\cal B}(D_{d}^{{\ast}-}{\to}{\ell}^{-}{\nu}_{\ell})$ ${\sim}$ ${\cal O}(10^{-10})$, ${\cal B}(B_{u}^{{\ast}-}{\to}{\ell}^{-}{\nu}_{\ell})$ ${\sim}$ ${\cal O}(10^{-10})$, ${\cal B}(D_{s}^{{\ast}-}{\to}{\ell}^{-}{\nu}_{\ell})$ ${\sim}$ ${\cal O}(10^{-6})$ and ${\cal B}(B_{c}^{{\ast}-}{\to}{\ell}^{-}{\nu}_{\ell})$ ${\sim}$ ${\cal O}(10^{-6})$. Inspired by the potential prospects of LHCb, Belle-II, STCF, CEPC and FCC-ee experiments, we discussed the probabilities of experimental investigation on these purely leptonic decays. It is found that the measurements of these decays might be possible and feasible with the improvement of data statistics, analytical technique, and measurement precision in the future. (1) With the hadron-hadron collisions, the purely leptonic decays of ${\rho}^{-}$, $K^{{\ast}-}$, $D_{d,s}^{{\ast}-}$ and $B_{u,c}^{{\ast}-}$ mesons might be accessible at LHC experiments. (2) With the $e^{+}e^{-}$ collisions, the purely leptonic decays of $D_{d,s}^{{\ast}-}$ and $B_{u,c}^{{\ast}-}$ mesons might be measurable with over $10^{12}$ $Z^{0}$ bosons available at CEPC and FCC-ee experiments. In addition, the $D_{d,s}^{{\ast}-}$ ${\to}$ ${\ell}^{-}{\nu}_{\ell}$ decays could also be studied at Belle-II and SCTF experiments.


I. INTRODUCTION
In the quark model [1][2][3], mesons are generally regarded as bound states of the valence quark q and antiquarkq . The classifications of mesons are usually based on the spin-parity quantum number J P of the qq system. The spin J of meson is given by the relation |L−S| ≤ J ≤ |L+S|. The orbital angular momentum and total spin of the qq system are respectively L and S, where S = 0 for antiparallel quark spins, and S = 1 for parallel quark spins. By Phenomenologically, by integrating out the contributions from heavy dynamical degrees of freedom such as the W fields, PLDCM can be properly described by the low-energy effective theory in analogy with the Fermi theory for β decays. Considering the fact that leptons are free from the strong interactions, the corresponding effective Hamiltonian [4] for PLDCM could be written as the product of quark currents and leptonic currents, where the contributions of the W bosons are embodied in the Fermi coupling constant G F 1.166×10 −5 GeV −2 [1], and V q 1 q 2 is the Cabibbo-Kobayashi-Maskawa (CKM) [5,6] matrix element between the quarks in the charged mesons. The decay amplitudes can be written as, The leptonic part of amplitudes can be calculated reliably with the perturbative theory. The hadronic matrix elements (HMEs) interpolating the diquark currents between the mesons concerned and the vacuum can be parameterized by the decay constants.
With the conventions of Refs. [7,8], the HMEs of diquark currents are defined as, where the nonperturbative parameters of f P and f V are the decay constants of pseudoscalar P and vector V mesons, respectively; and m V and µ are the mass and polarization vector, respectively. To the lowest order, the decay widths are written as, where m P and m are the masses of the charged pseudoscalar meson and lepton, respectively.
It is clearly seen from the above formula that the highly precise measurements of PLDCM will allow the relatively accurate determinations of the product of the decay constants and CKM elements, |V q 1 q 2 |f P,V . Theoretically, the decay constants are nonperturbative parameters, and they are closely related with theq 1 q 2 wave functions at the origin which cannot be computed from first principles. There still exist some discrepancies among theoretical results of the decay constants with different methods, such as the potential model, QCD sum rules, lattice QCD, and so on. If the magnitudes of CKM element |V q 1 q 2 | are fixed to the values of Ref. [1], the decay constants f P,V will be experimentally measured, and be used to seriously examine the different calculations on the decay constants with various theoretical models. Likewise, if the decay constants f P,V are well known to sufficient precision, the magnitudes of the corresponding CKM element will be experimentally determined, and provide complementary information to those from other processes. Within SM, the P → ν and V By considering the angular momentum conservation and the final states including a lefthanded neutrino or right-handed antineutrino, the purely leptonic decay width of charged pseudoscalar meson, Eq. (7), is proportional to the square of the lepton mass. This is called helicity suppression. While there is no helicity suppression for the purely leptonic decay of charged vector meson (PLDCV). From the analytical expressions of Eq.(7) and Eq.(8), the decay width of pseudoscalar meson is suppressed by the factor m 2 /m 2 P compared with that of vector meson. What's more, both the masses and the decay constants of vector mesons are relatively larger than those of corresponding pseudoscalar mesons, which would result in an enhancement of the decay widths for vector mesons. Of course, the vector mesons decay dominantly through the strong and/or electromagnetic interactions. The branching ratios for the PLDCV weak decays are usually very small, sometimes even close to the accessible limits of the existing and the coming experiments.
Inspired by the potential prospects of the future high-intensity and high-energy frontiers, along with the noticeable increase of experimental data statistics, the remarkable improvement of analytical technique and the continuous enhancement of measurement precision, the carefully experimental study of PLDCV might be possible and feasible. In this paper, we will focus on the PLDCV within SM to just provide a ready reference. The review of the purely leptonic decays of charged pseudoscalar mesons can be found in Ref. [1].
The mass of the ρ ± meson, m ρ = 775.11 (34) MeV [1], is much larger than that of two-pion pair. The rate of the ρ meson decay into two pions via the strong interactions is almost 100%, which results in the very short lifetime τ ρ ∼ 4.4×10 −24 s [1]. The direct measurements of the electroweak properties of the ρ meson would definitely be very challenging. It is evident from Eq.(8) that the parameter of |V ud | f ρ could be experimentally determined from the observations of decay widths for the ρ − → −ν decays (if it is not specified, the corresponding charge-conjugation processes are included in this paper), with the coupling constant G F , the masses of lepton m and meson m ρ .
The precise values of the CKM element |V ud | in ascending order of measurement accuracy mainly come from β transitions between the super-allowed nuclear analog states with quantum number of both J P = 0 + and isospin I = 1, between mirror nuclei with I = 1/2, between neutron and proton, between charged and neutral pions [9]. These four results for |V ud | are basically consistent with one another. The result of the super-allowed 0 + → 0 + nuclear β transitions has an uncertainty a factor of about 10 smaller than the other results, and thus dominates the weighted average value [9]. The best value from super-allowed nuclear β transitions is |V ud | = 0.97370 (14) [1], which is smaller compared with the 2018 value |V ud | = 0.97420(21) [10], as illustrated Fig. 1. This reduction of the value of |V ud | leads to a slight deviation from the first row unitarity requirement |V ud | 2 + |V us | 2 + |V ub | 2 = 1. The current precision of the CKM element |V ud | is about 0.01%. The latest value from the global fit in SM, |V ud | = 0.97401 (11) [1], will be used in our calculation.
The decay constant f ρ is an very important characteristics of the ρ meson. Compared with the CKM element |V ud |, the present precision of decay constant f ρ is still not very high and needs to be improved. Theoretically, the estimations from different methods are more or less different from each other and even calculations with the same method sometimes give the diverse results. Some theoretical estimations on the decay constant f ρ are presented in τ ± → ρ ± ν τ decay. The partial width for the τ → V ν τ decay is given by Ref. [36], where the factor S = 1.0154 includes the electroweak corrections [36][37][38]. With the mass MeV [33]. The more accurate decay constant f exp ρ will be used in our calculation.
b With light-cone wave functions and parameters of Ref. [13]. c With Gaussian wave functions and parameters of Ref. [12].
d With Gaussian wave functions and parameters of Ref. [13]. e With nonlocal condensates like functions.
h With Gaussian spatial wave functions and adjusted parameters of Ref. [20]. i With Gaussian spatial wave functions and parameters of Ref. [21]. j With rational spatial wave functions and adjusted parameters of Ref. [20].
k With rational spatial wave functions and parameters of Ref. [21]. ratios are very small. Given the identification efficiency and pollution from background, the ρ − → −ν decays might be measured only with more than 10 14 ρ ± events available.
There are at least three possible ways to experimentally produce the charged ρ mesons in the electron-position collisions, (a) the prompt pair production e + e − → ρ + ρ − , (b) the pair production via V decay 1 −− → ρ + ρ − , and (c) the single production via V decay 1 −− → ρ ± h ∓ . The cross section σ(e + e − →ρ + ρ − ) has been determined by the BaBar group to be 19.5±1.6±3.2 fb near the center-of-mass energy √ s = 10.58 GeV [39]. Assuming the production cross section σ ∝ 1/s [40,41], it could be speculated that σ(e + e − →ρ + ρ − ) ∼ 230 fb near √ s = 3.1 GeV. There would be only about 10 6 ρ + ρ − pairs with a data sample of 50 ab −1 [42] near √ s ≈ m Υ(4S) at the Belle-II detector or a data sample of 10 ab −1 [43] near √ s ≈ m J/ψ with the future super-tau-charm factory like STCF or SCTF [44][45][46]. The charge ρ mesons can in principle be produced from the Υ(4S), J/ψ and φ decays. The branching ratios are where the brancing ratio B(J/ψ→ρ + ρ − ) is assumed to be the same order of magnitude be accumulated. It is clearly seen that unless a very significant enhancement to branching ratios from some NP, the experimental data on the ρ ± meson are too scarce to search for the ρ − → −ν decays at the electron-position collisions in the near future, which result in the natural difficulties to understand the ρ meson.
The production cross sections of prompt J/ψ and J/ψ-from-b mesons in proton-proton collisions at √ s = 13 TeV are measured by LHCb to be 15.0±0.6±0.7 µb and 2.25±0.09±0.10 µb, respectively, [50]. It is expected that some 10 12 J/ψ events could be accumulated at √ s = 13 TeV with an integrated luminosity of 300 fb −1 at LHCb [51]. There are only about 10 10 ρ ± mesons from J/ψ decays available for prying into the ρ ± PLDCV decays. At the same time, the inclusive cross-sections for prompt charm production at LHCb at √ s = 13 TeV are measured to be O(1 mb) [52]. Analogically assuming the inclusive cross section of prompt ρ ± meson production at LHCb at √ s = 13 TeV is O(10 mb), some 3×10 15 ρ ± events would be accumulated with an integrated luminosity of 300 fb −1 at LHCb [51]. Optimistically assuming the reconstruction efficiency is about 10 %, there would be about O(10 2 ) events of the ρ − → −ν decays at LHCb, and more events with the enhanced branching ratios from NP contributions. Even through it will be very challenging for experimental analysis due to the complex background in hadron-hadron collisions, there is still a strong presumption that the ρ − → −ν decays could be explored and studied at LHC in the future. In addition, it is expected that an integrated luminosity exceeding 10 ab −1 would be reached at the future HE-LHC experiments [53]. More experimental data at HE-LHC would make the study of the ρ − → −ν decays indeed feasible in hadron-hadron collisions.
The parameter product |V us | f K * could be experimentally determined from the K * − → −ν decays using Eq. (8). Like the ρ ± meson, the mass of the K * ± meson, m K * ± = 895.5 (8) MeV, is above the threshold of Kπ pair, and the partial branching ratio of the K * meson decay into Kπ pair via the strong interactions is almost 100% [1]. It is not hard to imagine that the very short lifetime τ K * ∼ 1.4×10 −23 s would enable the measurements of the electroweak properties of the K * meson to be very challenging or nearly impossible.
The CKM element |V us | λ up to the order of O(λ 6 ), where λ is a Wolfenstein parameter.
The current precision of the CKM element |V us | from purely leptonic and semileptonic K meson decays and hadronic τ decays are 0.2%, 0.3% and 0.6%, respectively. It is seen from Some theoretical results on the decay constant f K * are presented in Table II. Like the case of the decay constant f ρ , the model dependence of theoretical estimations on the decay constant f K * is also obvious. Experimentally, the decay constant f K * can be obtained from the hadronic τ ± → K * ± ν τ decays. Using Eq.(9) and experimental data on branching ratio The value of f exp K * is much less than that of LQCD results, and will be used in our calculation. For the K * − → −ν decays, the SM expectations on the partial decay widths and branching ratios are, The decay width Γ K * = 46.2±1.3 MeV [1] is used in our calculation. It is apparent that more than 10 14 K * ± events are the minimum requirement for experimentally studying the Based on the U -spin symmetry, the production mechanism of the K * ± mesons in electronposition collisions is similar to that of the ρ ± mesons. An educated guess is that the cross section σ(e + e − →K * + K * − ) ∼ 20 fb and 230 fb near √ s ∼ m Υ(4S) and m J/ψ , respectively.
b With Gaussian spatial wave functions and parameters of Ref. [21]. c With rational spatial wave functions and adjusted parameters of Ref. [20].
d With rational spatial wave functions and parameters of Ref. [21]. e With Coulomb plus linear potential model.
f With Coulomb plus harmonic oscillator potential model. g With a dilation parameter κ = 0.54 GeV.
h With a dilation parameter κ = 0.68 GeV. i The K meson mass is used as input. j The φ meson mass is used as input.
It is approximately estimated that B(J/ψ→K * ± X ∓ ) ∼ 1.8 %. Hence, the experiemtal data on the K * ± mesons at the e + e − collisions, which would be available by either the prompt K * + K * − pair production at SuperKEKB and SCTF experiments or the production via 10 13 J/ψ decay at SCTF, are far from sufficient for investigating the K * − → −ν decays. If we assume that the inclusive cross section of prompt K * ± meson production in pp collisions at the center-of-mass energy of 13 TeV is similar to that of ρ ± mesons, about O(10 mb), there would be some 3×10 15 K * ± events to be available with an integrated luminosity of 300 fb −1 at LHCb, which correspond to about O(10 2 ) events of the K * − → −ν decays. It should be some glimmer of hope for observation and scrutinies of the K * − → −ν decays at hadron-hadron collisions in the future, particularly at the planning HE-HLC.
The mass of D * d mesons, m D * d = 2010.26(5) MeV, are just above the threshold of Dπ pair. The D * d meson decays via the strong interactions are dominant, and the ratio of branching ratios [1], B(D * ± d →D ± d π 0 )/B(D * ± d →D u π ± ) = 30.7(5) %/67.7(5) % ∼ 1/2, basically agrees with the relations of isospin symmetry. It should be pointed out that the D * d strong decays are highly suppressed by the compact phase spaces because of m D * d − m D − m π < 6 MeV. The branching ratio of the magnetic dipole transition is small, Currently, the precise values of the CKM element |V cd | comes mainly from the leptonic and semileptonic D meson decays [1], as illustrated in Fig. 3. Because of the decay width of Eq. (7) being proportional to m 2 , the D − → e −ν e decay is helicity suppressed. And the D − → τ −ν τ decay suffers from the complications caused by the additional neutrino in τ decays. k With a dilation parameter κ = 0.68 GeV.
The D − → µ −ν µ decay is the most favorable mode for experimental measurement. For the values of |V cd | from the purely leptonic decay D − → µ −ν , the experimentally statistical uncertainties are dominant uncertainties. For the values of |V cd | from the semileptonic D meson decays, the theoretical uncertainties from the form factor controlled by nonperturbative dynamics are dominant uncertainties. It is clearly seen from Fig. 3 that the experimental uncertainties have not decreased significantly recently. Besides, |V cd | can also be determined from the neutrino-induced charm production data [1], but the relevant experimental data have not been updated after the measurements given by the CHARM-II Collaboration in 1999 [55]. According to the Wolfenstein parameterization of the CKM matrix, there is an approximate relation between its elements |V cd | = |V us | = λ up to O(λ 4 ). However, the measurement precision of the CKM element |V cd | from both leptonic and semileptonic D meson decays is generally about an order of magnitude smaller than that of |V us | from leptonic and  Table III. The theoretical discrepancies among various methods are obvious. In our calculation, as a conservative estimate, we will take the recent value f D * d = 230±29 MeV [28] from the light front quark model, which agrees basically with the values f D * d = 234±6 MeV [73] from the recent lattice QCD simulation. After some simple computation with Eq. (8), we obtain the partial decay widths and branching ratios for the D * − d → −ν decays as follows.  lations of the form factors. It is wroth noting that the recent CKM element |V cs | determined by the BES-III group from the D + s → µ + ν µ and D + s → τ + ν τ decays based on available 6.32 fb −1 data is |V cs | = 0.978±0.009±0.014 [94], where the systematic (second) uncertainties has outweighed the statistical (first) one. This value is very close to the precise result from the global fit, |V cs | = 0.97320 (11) [1] that will be used in this paper.
By now, a relatively little information about the properties of the D * s mesons is available. For example, the quantum number of J P , the decay constant f D * s , and the width Γ D * s have not yet been determined or confirmed explicitly by experiments. It is generally thought that the J P of the D * s mesons is consistent with 1 − from decay modes [96]. Some theoretical results on the decay constant f D * s are listed in Table IV. It can be seen that the theoretical results are various. The recent LQCD results on the decay constant from ETM [72], HPQCD [97] and χQCD [73] groups are in reasonable agreement with each other within an error range. k With a dilation parameter κ = 0.68 GeV.
The latest decay constant f D * s = 274±7 MeV from LQCD calculation [73] will be used for an estimation for PLDCV of the D * s mesons in this paper. The experimental upper limit of the decay width is Γ D * s < 1.9 MeV at the 90 % confidence level set by the CLEO collaboration in 1995 [96]. An approximate relation for the decay width, Γ D * s Γ(D * s →γD s ), is often used in theoretical calculation. The radiative transition process, D * s → γD s , is a parity conserving decay. The parity and angular momentum conservation implies that the orbital angular momentum of final states L = 1. There are many theoretical calculation on the decay width Γ D * s , for example, Refs. . The partial decay width for the magnetic dipole transition is generally written as [129], with the definition of the magnetic dipole moment µ V P and the momentum of photon k γ in the rest frame of the vector meson, where Q i and m i are the electric charge in the unit of |e| and mass of the constituent quark, respectively. With m d ≈ 336 MeV, m s ≈ 490 MeV, m c ≈ 1500 MeV and the one can obtain Γ(D * d →γD d ) ≈ 1.8 keV and Γ(D * s →γD s ) ≈ 0.36 keV [129]. The theoretical value of partial decay width Γ(D * d →γD d ) is roughly consistent with the corresponding experimental data Γ(D * d →γD d ) = Γ D * d ×B(D * d →γD d ) = 1.33±0.33 keV within 2 σ regions [1]. For the moment, we will use Γ D * s = 0.36 keV in the calculation to give an estimate of branching ratios for the D * − s → −ν decays.
If considering the experimental measurement efficiency, there are at least more than 10 7 D * s events to experimentally study the D * − d → −ν decays. And more than 10 8 D * s events might be needed to explore the D * − s → τ −ν τ decay. In the electron-positron collisions, the cross sections of D + s D * − s and D * + s D * − s production have been experimentally studied by the Belle [130], BaBar [131] and CLEO-c [84,85]  groups, as illustrated in Fig. 6. Assuming the exclusive cross sections near threshold σ(e + e − →D + s D * − s ) ∼ 1.0 nb and σ(e + e − →D * + s D * − s ) ∼ 0.2 nb, there will be about 10 10 D * ± s events corresponding to a data sample of 10 ab −1 at STCF, and about 5×10 10 D * ± s events corresponding to a data sample of 50 ab −1 at SuperKEKB. In addition, considering the branching ratio B(Z→cc) = (12.03±0.21) % [1] and the fragmentation fraction f (c→D * s ) 5.5 % [132], there will be more than 6×10 9 (and 6×10 10 ) D * ± s events corresponding to 10 12 [86] (and 10 13 [87]) Z bosons at the future CEPC (and FCC-ee). So the D * − s → −ν decays (with = e, µ and τ ) could be measured at Belle-II, SCTF, CEPC and FCC-ee experiments.
In hadron-hadron collisions, the inclusive cross sections for the cc pair production are σ(pp→ccX) 2.4 mb at the center-of-mass energy of √ s = 13 TeV at LHCb [52], σ tot cc 8.5 mb and 8.6 mb at √ s = 7 TeV at ALICE [88] and ATLAS [89], respectively. With the fragmentation fraction f (c→D * s ) 5.5 % [132], there will be about 4×10 13 D * ± s events corresponding a data sample of 300 fb −1 at LHCb, and more D * ± s events available at ALICE and ATLAS. So the D * − s → e −ν e , µ −ν µ , τ −ν τ decays could be measured precisely at LHCb, ALICE and ATLAS experiments.
The experimental information about the B * u mesons are very scarce. The already known information about the B * u mesons are their quark composition bū with the quark model assignment, the isospin I = 1/2, the spin-parity quantum number J P = 1 − and the mass m B * u = 5324.70(21) MeV [1]. Due to the mass difference m B * u − m Bu = 45 MeV < m π , the electromagnetic radiative transition B * u → B u γ certainly will be the important and dominant decay mode. The photon in the B * u → B u γ decay is very soft, with the momentum k γ ∼ 45 MeV in the center-of-mass of the B * u mesons. No signal event of the B * u → B u γ decay has yet been found. The B * − u → −ν decays offer a complementary decay modes of the B * u meson. It can be seen from Eq.(8) that the information about |V ub | f B * u could be obtained, however, the partial width for the purely leptonic decays B * − u → −ν are highly suppressed by the CKM element of |V ub | 2 ∼ O(λ 6 ).
The precise determinations of the CKM element V ub = |V ub | e −i γ are very central and important to verify the CKM picture of SM, where γ is the angle of the unitarity triangle |V ub | ×10 3 = 3.70±0.10 exp ±0.12 theo = 3.70±0.16 (exclusive).
one can obtain Γ B * u Γ(B * u →γB u ) 820 eV. The partial decay width and branching ratios for the B * − u → −ν decays are It is expected that there should be at least more than 10 11 B * u events available for experimental study of the B * − u → −ν decays. The experimental study has shown that the exclusive cross sections for the final states of BB * , B * B * and BB * π will have a large share of the total bb cross sections above the open bottom threshold, for example [1], There are about 36×10 6 Υ(5S) events corresponding to the dataset of 121 fb −1 at Belle experiments at the disposal [42]. About 1.5×10 10 Υ(5S) events with a dataset 50 ab −1 at Belle-II are an outside estimate. Assuming the inclusive branching ratio B(Υ(5S)→B * u X) 30 %, there will be some 4.5×10 9 B * u events at most at Belle-II. And it is more important that the vast majority of the data will be taken at Υ(4S) resonance rather than Υ(5S) 495 µb at LHCb [50]. There will be more than 5×10 13 B * u events with a dateset of 300 fb −1 at LHCb and fragmentation fraction f (b→B * u ) ∼ 20 %. Hence, the B * − u → e −ν e , µ −ν µ , τ −ν τ decays could be investigated at FCC-ee and LHCb experiments in the future.
According to the conventional quark-model assignments, the B * c mesons consist of two heavy quarks with different flavor numbers B = C = −Q = ±1. Up to today, the experimental information of the B * c meson is still very limited. For example, the potential candidate of the B * c meson has not yet been determined. It is generally believed that the mass of the B * c meson should be in the region between m Bc = 6274.47±0.27±0.17 MeV recently measured by LHCb [137] and m Bc(2S) = 6872.1±1.3±0.1±0.8 MeV obtained by LHCb [138] (or 6871.0±1.2±0.8±0.8 MeV given by CMS [139]), where the B c , B * c and B c (2S) particles correspond to the sibling isoscalar states with quantum numbers of n 2S+1 L J = 1 1 S 0 , 1 3 S 1 and 2 1 S 0 , respectively. So the branching ratios for the strong decays B * c → BD are zero, because B * c meson is below the BD pair threshold. The experimental particle physicists are earnestly looking for and identifying the B * c meson, a long-expected charming beauty. For the moment, almost all of the information available about the properties of B * c meson (such as the mass, decay constant, lifetime, decay modes and so on) come from theoretical estimates. There are too many estimations on the B * c meson mass with various theoretical models, for example, in Refs. . The recent result from lattice QCD calculation, [179], which are basically consistent with other estimations, will be used in this paper. Clearly, it is foreseeable that the isospin violating decay B * c → B c π is explicitly forbidden by the law of energy conservation, because of m B * c − m Bc 57 MeV < m π . Hence, the electromagnetic radiative transition B * c → B c γ should be the dominant decay mode. In addition, the photon in the magnetic dipole transition B * c → B c γ is very soft in the rest frame of the B * c meson. This might be one main reason why the unambiguously experimental identification of the B * c meson is very challenging. As an important complementary decay modes, the B * c meson has very rich weak decay channels, which could be approximately classified into three classes: (1) the valence b quark weak decay accompanied by the spectator c quark, (2) the valence c quark weak decay accompanied by the spectator respectively [1]. The lepton flavor non-universality in the ratio R(D ( * ) ) complicate the determination of |V cb |. In addition, |V cb | can also be obtained from the PLDCM B − c → ν decays, although none of the measurements has reached a competitive level of precision due to either the serious helicity suppression for B − c → eν e , µν µ decays or other additional neu- trinos from τ decay for B − c → τν τ decay. The global SM fit value is |V cb | = 40.53 +0.83 −0.61 ×10 −3 [1], which will be used in this paper.
Both valence quarks of the B ( * ) c mesons are regarded as heavy quarks. Their Compton wave lengths ∼ 1/m b,c are much shorter than a typical hadron size. The spin-flavor symmetry in the heavy quark limit would lead to an approximation between decay constants f Bc ≈ f B * c . Some theoretical results on the decay constant f B * c are collected in Table VI. The recent lattice QCD calculation f B * c = 387±12 MeV [197] will be used in this paper. As it is well known that the magnetic momentum of both b and c quarks are inversely proportional to their mass. The magnetic dipole momentum To experimentally investigate the B * − c → −ν decays, there should be at least more than 10 7 B * c events available. More than 10 12 Z bosons are expected at the future e + e − colliders of CEPC [86] and FCCee [87]. Considering the branching ratio B(Z→bb) = 12.03±0.21 % [1] and fragmentation fraction f (b→B * c ) ∼ 6×10 −4 [198][199][200], there will be more than 10 8 B * c events to search for the B * − c → e −ν e , µ −ν µ , τ −ν τ decays. In addition, the B * c production cross sections at LHC are estimated to be about 100 nb for pp collisions at √ s = 13 TeV, about 8 mb for p-Pb collisions at √ s = 8.16 TeV and some 920 mb for Pb-Pb collisions at √ s = 5.02 TeV, respectively [201]. There will be more than 3×10 10 B * c events corresponding to a dataset of 300 fb −1 at LHCb for pp collisions. Hence, the B * − c → e −ν e , µ −ν µ , τ −ν τ decays are expected to be carefully measured at LHCb experiments in the future.

VIII. SUMMARY
The mass of the charged vector mesons are generally larger than that of the corresponding ground pseudoscalar mesons. The vector mesons decay mainly through the strong or/and electromagnetic interactions. These facts will inevitably result in that the branching ratios of the vector meson weak decays are often very tiny. Inspired by the potential prospects of existing and coming high-luminosity experiments, more and more experimental data will be accumulated, and higher measurement precision level will be reached. The probabilities of experimental investigation on the purely leptonic decays of charged vector mesons are discussed in this paper. We found that (1) for both ρ ± and K * ± mesons, their widths are large due to the dominance of strong decay. Their PLDCV branching ratios are estimated at the order of O(10 −13 ). Although extremely complicated and difficult, the PLDCV decays ρ ± , K * ± → e −ν e , µ −ν µ might be measurable due to the huge data of the ρ ± and K * ± mesons at LHCb. (2) The PLDCV D * s decays are favored by the CKM element |V cs |. Their branching ratios are about O(10 −6 ). The PLDCV decays D * d,s → e −ν e , µ −ν µ , τ −ν τ could be carefully studied at the Belle-II, SCTF or STCF, CEPC, FCC-ee, LHCb experiments. (3) For the B * u mesons below the Bπ thresholds and the B * c mesons below both BD and B c π thresholds, they decay predominantly through the magnetic dipole transitions. The branching ratios of the PLDCV B * c decays favored by the CKM element |V cb | could reach up to O(10 −6 ). The PLDCV decays B * u,c → e −ν e , µ −ν µ , τ −ν τ might be searched for at the CEPC, FCC-ee, LHCb experiments. Our rough estimations and findings are summed in Table VII. We wish that our investigation could provoke physicists' researching interest in PLDCV and offer a ready reference for the future experimental analysis.